## Large Sample Properties of Generalized Estimating Equations with Adaptive Designs for Longitudinal Data

Yin Changming,, Shi Yuexin

School of Mathematics and Information Science, Guangxi University, Nanning 530004

 基金资助: 国家自然科学基金.  11061002国家自然科学基金.  11701109广西自然科学基金.  2015GXNSFAA139006广西自然科学基金.  2018GXN-SFBA281016

 Fund supported: the NSFC.  11061002the NSFC.  11701109the NSF of Guangxi.  2015GXNSFAA139006the NSF of Guangxi.  2018GXN-SFBA281016

Abstract

Generalized estimating equation (GEE) is widely adopted in analyzing longitudinal (clustered) data with discrete or nonnegative responses. In this paper, we prove the existence, weak consistency and asymptotic normality of generalized estimating equations estimator with adaptive designs under some mild regular conditions. The accuracy of the asymptotic approximation is examined via numerical simulations. Our results extend the elegant work of Xie and Yang (Ann Statist, 2003, 31: 310-347) and Balan and Schiopu-Kratina (Ann Statist, 2005, 33: 522-541).

Keywords： Generalized estimating equations ; Adaptive designs ; Longitudinal data ; Asymptotic normality

Yin Changming, Shi Yuexin. Large Sample Properties of Generalized Estimating Equations with Adaptive Designs for Longitudinal Data. Acta Mathematica Scientia[J], 2021, 41(6): 1925-1936 doi:

## 1 引言

$Y_{ij}$是第$i$个个体的第$j$次观测响应变量的值, 回归因子$X_{ij} $$p\times 1 设计阵, i = 1, 2, \cdots , n , j = 1, 2, \cdots , m_i . Y_i = (Y_{i1}, \cdots , Y_{im_i})^\tau , X_i = (X_{i1}, \cdots , X_{im_i}) , 单调递增的 \sigma 代数 {{\cal F}}_{i}: = \sigma\{(Y_k, X_k), k = 1, \cdots, i\} . X_{i}$$ {\cal F}_{i-1}$可测, 即$X_{i}$的设计依赖于前面的观察值, 称为自适应的, 且假设

$$$\mu_{ij}(\beta): = \mbox{E}(Y_{ij}|{{\cal F}}_{i-1}) = \mu(X_{ij}^\tau\beta),$$$

$$$\sigma_{ij}(\beta): = \mbox{Var}(Y_{ij}|{{\cal F}}_{i-1}) = \dot\mu(X_{ij}^\tau\beta),$$$

$\begin{eqnarray} g_n(\beta) = \sum\limits_{i = 1}^nX_iA_i^{1/2}(\beta) R_i^{-1} A_i^{-1/2}(\beta)(Y_i-\mu_i(\beta)) = 0, \end{eqnarray}$

GEE是广义线性模型(GLM)的推广, GLM和GEE的建模可参看文献[1, 4-7]. GLM大样本性质的可参看文献[8-13]. GEE大样本性质的可参看文献[2, 3, 14]. 文献[2]严格证明了工作相关阵已知, 每个个体观测次数可以无界情形下GEE估计的大样本性质. 文献[14]证明了协变量的维数可以无界情形下GEE估计的渐近性质. 文献[3]证明了具有估计工作相关阵, 误差是鞅差情形GEE估计的相合性和渐近正态性. 文献[2, 14]都假设$Y_i, i\ge 1$是相互独立的, $X_{ij}$是固定的. 文献[3]假设$\mbox{E}(Y_{ij}|{{\cal F}}_{i-1}) = \mu(X_{ij}^\tau\beta)$, 而设计阵$X_{ij}$也还是假设是固定的, 非随机的. 自适应设计在心里学研究, 控制理论有广泛地应用, 可参见文献[6, 7, 15, 16]. 该文在较弱的条件下证明了自适应设计GEE回归参数估计的渐近性质, 并做了数值模拟. 把文献[2]和[3]的相应结果推广到了设计阵是自适应情形, 并且对Fisher信息阵的特征根的要求降到了最低.

## 2 主要结果

(A1) (ⅰ) $\|X\|_n^2 / \lambda_{\min}(F_n)\to_{a.s} 0,$ (ⅱ) $\sup\limits_{i\ge 1}m_i<\infty$, 其中Fisher信息阵$F_n = $$\sum\limits_{i = 1}^n X_iX_i^\tau , \|X\|_n = \max\limits_{ i\le n, j\le m_i} \|X_{ij}\|^2 , 对一个矩阵 T , \|T\| = [\mbox{trace}(TT^\tau)]^{1/2} , \lambda_{\min}(T)$$ \lambda_{\max}(T)$分别表示矩阵$T$的最小和最大特征值.

(A2) 未知参数$\beta$属于紧集${\mathbb B}\subset {{\Bbb R}} ^{p}$, 参数真值$\beta_{0} $$\mathbb B 的内点. (A3) 存在一个常数 \alpha_0>2 使得 其中 Y_{ij}^* = Y_{ij}^*(\beta_0), Y_{ij}^*(\beta) = A_{ij}^{-1/2}(\beta)(Y_{ij}-\mu_{ij}(\beta)) . (A4) 概率为1, \inf\limits_{i\ge1}\lambda_{\min}( R_i)>0 , \inf\limits_{i\ge1}\lambda_{\min}(\bar R_i)>0 , 其中 \bar R_i = E(Y_i^*(Y_i^*)^\tau|{\cal F}_{i-1}) 是真实相关阵, Y_{i}^* = Y_{i}^*(\beta_0), Y_{i}^*(\beta) = A_{i}^{-1/2}(\beta)(Y_{i}-\mu_{i}(\beta)). (A5) 存在一个非随机矩阵 \bar M_n 使得 \bar M_n^{-1}M_n^{1/2}\to_p I_p , 其中 I_p$$ p$阶单位阵, $M_n = M_n(\beta_0), $$M_n(\beta) = \sum\limits_{i = 1}^n X_iA_i^{1/2} R_i^{-1}\bar R_i R_i^{-1}A_i^{1/2}X_i^\tau . (A6) (ⅰ) \sup\limits_{n\ge 1}\sup\limits_{i\le n, j\le m_i, \beta\in \tilde B_n(r)} |\mu^{(k)}(X_{ij}^\tau\beta)|<\infty \; \; a.s. , \mu^{(k)} 表示 \mu$$ k$阶导数, $k = 1, 2, 3$. (ⅱ) $\inf\limits_{n\ge 1}\inf\limits_{i\le n, j\le m_i, \beta\in \tilde B_n(r)}\dot \mu(X_{ij}^\tau\beta)>0\; \; a.s.,$其中$\tilde B_n(r) = \{\beta :\|\bar M_n^\tau(\beta-\beta_0)\|\le r\}$, $r$是任意正常数.

$$$P( g_n(\hat\beta_n) = 0)\to 1, \qquad \hat\beta_n\to_p\beta_{0}, \nonumber$$$

$$$M_n^{-1/2} H_n(\hat\beta_n-\beta_{0})\to_d N(0, I_p), \nonumber$$$

## 3 主要结果的证明

$\begin{eqnarray} \sup\limits_{\beta\in \tilde B_n(r)}\|H_{n}^{-1/2}D_{n}(\beta)H_{n}^{-1/2}-I\| = o_p(1). \end{eqnarray}$

$\begin{eqnarray} \sup\limits_{\beta\in \tilde B_n(r)}\|H_{n}^{-1/2}D^*_{n}(\beta)H_{n}^{-1/2}-I\| = o_p(1), \end{eqnarray}$

$\begin{eqnarray} \sup\limits_{\beta\in \tilde B_n(r)}\left\|\left[H_{n}^{-1/2}D^*_{n}(\beta)H_{n}^{-1/2}\right]^{-1}-I\right\| = o_p(1). \end{eqnarray}$

$\begin{eqnarray} P(CI\le R_i\le C_1\bar R_i\le C_2A_i\le C_3I, i\ge 1)>1-\epsilon, \end{eqnarray}$

$\begin{eqnarray} P(F_n\le CH_n\le C_1M_n\le C_2\bar M_n\bar M_n^\tau\le C_3F_n)>1-\epsilon. \end{eqnarray}$

$\begin{eqnarray} \|H_{n}^{-1/2}g_{n}\| = \|H_{n}^{-1/2}M_{n}^{1/2}\times M_{n}^{-1/2}g_{n}\| = O_p(1). \end{eqnarray}$

$\partial \tilde B_n(r) = \{\beta :\|\bar M_n^\tau(\beta-\beta_0)\| = r\}$. 由式(3.5)知, 对任意$\epsilon>0$, 存在$C_5>0$使得

$\begin{eqnarray} P\Big(\inf\limits_{\beta\in \partial \tilde B_n(r)}(\beta-\beta_0)^\tau H_n(\beta-\beta_0)\ge C_5r^2\Big)>1-\epsilon, \end{eqnarray}$

$\begin{eqnarray} P\Big(\inf\limits_{\|\lambda\| = 1, \beta\in \tilde B_n(r)}\lambda^\tau H^{-1/2}_{n}D^*_{n}(\beta)H^{-1/2}_{n}\lambda\ge \frac{1}{2}\Big)>1-\epsilon. \end{eqnarray}$

$\begin{eqnarray} &&(\beta-\beta_0)^\tau g_{n}(\beta) = (\beta-\beta_0)^\tau g_{n}-(\beta-\beta_0)^\tau[g_{n}-g_{n}(\beta)] \\ & = &(\beta-\beta_0)^\tau g_{n}-(\beta-\beta_0)^\tau D^*_{n}(\beta)(\beta-\beta_0)\\ & = &(\beta-\beta_0)^\tau\bar M_{n}\times\bar M^{-1}_{n} M^{1/2}_{n}\times M^{-1/2}_{n}g_{n}\\ &&-(\beta-\beta_0)^\tau H^{1/2}_{n}\left[ H^{-1/2}_{n}D^*_{n}(\beta)H^{-1/2}_{n}\right] H^{1/2}_{n}(\beta-\beta_0)\\ &\le&C r-\frac{C_5}{2}r^2. \end{eqnarray}$

$\begin{eqnarray} \mbox{P} \Big\{ \sup\limits_{\beta\in \partial \tilde B_n(r)}(\beta-\beta_0)^\tau g_{n}(\beta)<0\Big\}\to 1. \end{eqnarray}$

$\mbox{P} \Big\{ \mbox{存在 }\ \hat\beta_{n}\in \tilde B_n(r)\ \mbox{ 使得 }\ g_{n}(\hat\beta_{n}) = 0\Big\}\to 1,$

$\begin{eqnarray} \|\hat\beta_{n}-\beta_0\| = \|F^{-1/2}_{n}.F^{1/2}_{n}(\bar M^\tau_{n})^{-1}.\bar M^\tau_{n}(\hat\beta_{n}-\beta_0)\| = o_p(1). \end{eqnarray}$

$\begin{eqnarray} g_{n}(\beta_0) = D_{n}^*(\hat\beta_{n})(\hat\beta_{n}-\beta_0). \end{eqnarray}$

$\begin{eqnarray} &&M_{n}^{-1/2}H_{n}(\hat\beta_{n}-\beta_0) \\& = & M_{n}^{-1/2} H_{n}^{1/2}\left\{[H_{n}^{-1/2}D^*_{n}(\hat\beta_{n})H_{n}^{-1/2}]^{-1}-I\right\}[M_{n}^{-1/2} H_{n}^{1/2}]^{-1}M_{n}^{-1/2}g_{n}(\beta_0) \\ &&+M_{n}^{-1/2}g_{n}(\beta_0). \end{eqnarray}$

$\begin{eqnarray} M_{n}^{-1/2}H_{n}(\hat\beta_{n}-\beta_0)\to_d N(0, I). \end{eqnarray}$

$\beta_0 = (\beta_{00}, \beta_{01})^\tau$, $X_{ij} = (1, Y_{i-1, j})^\tau$, 其中$Y_{0j} = 0$. 首先我们取$m = 3$, $\beta_0 = (-0.9, 0.4)^\tau$, $Y_i = (Y_{i1}, Y_{i2}, Y_{i3})^\tau$有相关系数为0.3的交换相关结构(即$\bar R_{i} = 0.3{\bf\bar 1_3}+0.7I_3$, 其中$\bf {\bar 1_3}$是元素都为1的$3\times 3$矩阵, $I_3 $$3 阶单位阵). 表 1给出了样本容量分别为 n = 50, 100, 200, 400 , 工作相关阵分别取真实相关阵( R_i = \bar R_{i} ), 独立相关阵( R_i = I ), 一阶自回归相关阵(即 R_i 的第 j 行, j' 列的元素为 0.3^{|j-j'|} ) 的模拟结果. 表 1 m_i = 3, i\ge 1时估计的均值(标准差)  \beta_{01} = 0.4 \beta_{00} = -0.9 真实相关阵 独立相关阵 自回归相关阵 真实相关阵 独立相关阵 自回归相关阵 n = 50 0.3681 0.3474 0.3613 -0.9067 -0.9014 -0.9053 (0.3861) (0.4132) (0.3975) (0.2713) (0.2741) (0.2748) n = 100 0.3990 0.3716 0.3747 -0.9057 -0.9040 -0.9055 (0.2576) (0.2800) (0.2639) (0.1830) (0.1845) (0.1833) n = 200 0.3862 0.3837 0.3856 -0.9051 -0.9047 -0.9056 (0.1830) (0.1971) (0.1868) (0.1249) (0.1249) (0.1244) n = 400 0.3926 0.3917 0.3920 -0.8988 -0.8987 -0.8987 (0.1316) (0.1411) (0.1340) (0.0898) (0.0911) (0.0903) 然后我们取 m = 8 , \beta_0 = (0.9, 0.4) , Y_i = (Y_{i1}, Y_{i2}, \cdots, Y_{i8})^\tau 有相关系数为0.3的交换相关结构. 表 2给出了样本容量分别为 n = 50, 100, 200, 400 , 三种工作相关阵分别为真实相关阵, 独立(工作)相关阵, 一阶自回归(工作)相关阵的模拟结果. 表 2 m_i = 8, i\ge 1时估计的均值(标准差)  \beta_{01} = 0.4 \beta_{00} = 0.9 真实相关阵 独立相关阵 自回归相关阵 真实相关阵 独立相关阵 自回归相关阵 n = 50 0.3761 0.3480 0.3667 0.9358 0.9596 0.9437 (0.2814) (0.3785) (0.3124) (0.2873) (0.3485) (0.3098) n = 100 0.3928 0.3778 0.3868 0.9128 0.9254 0.9178 (0.1886) (0.2460) (0.2031) (0.1955) (0.2338) (0.2069) n = 200 0.4047 0.4018 0.4034 0.9030 0.9089 0.9045 (0.1288) (0.1756) (0.1427) (0.1371) (0.1648) (0.1472) n = 400 0.4031 0.3961 0.4015 0.8997 0.9021 0.9006 (0.0930) (0.1236) (0.1039) (0.0938) (0.1127) (0.1007) 表 1表 2可看出随着样本容量 n 的增加, 回归参数估计的均值越来越接近真值, 偏差(标准差)越来越小. 而且当工作相关阵等于真实相关阵时, 估计的标准差最小. ## 5 引理3.4至引理3.7的证明 引理3.4的证明 由假设(A1)(ⅰ), (3.5)式和 \tilde B_n(r) 的定义知 $$\sup\limits_{i\le n, j, \beta\in \tilde B_n(r)}\| X_{ij}^\tau(\beta-\beta_0)\|\le \sup\limits_{i\le n, j, \beta\in \tilde B_n(r)} \| X_{ij}^\tau F_n^{-1/2}.(\bar M_n^{-1} F_n^{1/2})^\tau.\bar M_n^\tau(\beta-\beta_0)\| = o_p(1).$$ 由微分中值定理, 假设(A6)(ⅰ), 式(5.1)知 \begin{eqnarray} \sup\limits_{i\le n, j, \beta\in \tilde B_n(r)}|A_{ij}(\beta)-A_{ij}| = |\mu^{(2)}(X_{ij}^\tau\beta^*)|\| X_{ij}^\tau(\beta-\beta_0)\| = o_p(1), \end{eqnarray} 其中 \beta^*$$ \beta_0 $$\beta 的连线上. 由(A6)(ⅱ), 式(5.2)知 \begin{eqnarray} \sup\limits_{i\le n, j, \beta\in \tilde B_n(r)}|A^{1/2}_{ij}(\beta)-A^{1/2}_{ij}| = \frac{|A_{ij}(\beta)-A_{ij}|}{A^{1/2}_{ij}(\beta)+A^{1/2}_{ij}} = o_p(1). \end{eqnarray} \lambda_k$$ \lambda_l$分别是第$k$个和第$l$个分量为1, 其余元素都为0的$p$维单位向量, $1\le k, l\le p$. 由式(3.5)知

$\begin{eqnarray} \sum\limits_{i = 1}^n\lambda_{l}^\tau H^{-1/2}_nX_i X^\tau_i H^{-1/2}_n\lambda_{ l} = O_p(1). \end{eqnarray}$

$\begin{eqnarray} &&\sup\limits_{i\le n, \beta\in \tilde B_n(r)}\bigg(\sum\limits_{i = 1}^n\lambda_{k}^\tau H^{-1/2}_nX_i[ A^{ 1/2}_i(\beta)- A^{ 1/2}_i] R_i^{-1} A^{ 1/2}_i(\beta) X^\tau_i H^{-1/2}_n\lambda_{ l}\bigg)^2\\ &\le& \sup\limits_{i\le n, \beta\in \tilde B_n(r)}\sum\limits_{i = 1}^n\lambda_{ k}^\tau H^{-1/2}_nX_i[ A^{ 1/2}_i(\beta)- A^{ 1/2}_i]^2 X^\tau_i H^{-1/2}_n\lambda_{ k}\\ &&\times\sup\limits_{i\le n, \beta\in \tilde B_n(r)}\sum\limits_{i = 1}^n \lambda_{ l}^\tau H^{-1/2}_nX_i A^{ 1/2}_i(\beta)R_i^{-2}A^{ 1/2}_i(\beta) X^\tau_i H^{-1/2}_n\lambda_{ l}\\ &\le &\sup\limits_{i\le n, \beta\in \tilde B_n(r)} \lambda_{\max}\left\{[ A^{ 1/2}_i(\beta)- A^{ 1/2}_i ]^2 \right\}\sum\limits_{i = 1}^n\lambda_{k}^\tau H^{-1/2}_nX_i X^\tau_i H^{-1/2}_n\lambda_{ k}\\ &&\times \sup\limits_{i\ge 1, \beta\in \tilde B_n(r)}\lambda_{\max}(R_i^{-2} A_i (\beta))\sum\limits_{i = 1}^n\lambda_{l}^\tau H^{-1/2}_nX_i X^\tau_i H^{-1/2}_n\lambda_{ l}\\ & = &o_p(1). \end{eqnarray}$

$\begin{eqnarray} \bigg( \sum\limits_{i = 1}^n\lambda_{ k}^\tau H^{-1/2}_nX_i A^{ 1/2}_iR_i^{-1}[A^{ 1/2}_i(\beta) -A^{ 1/2}_i ] X^\tau_i H^{-1/2}_n\lambda_{ l}\bigg)^2 = o_p(1). \end{eqnarray}$

$\begin{eqnarray} \sup\limits_{\beta\in \tilde B_n(r)}|\lambda_{ k}^\tau H^{-1/2}_n[H_n(\beta)-H_n] H^{-1/2}_n\lambda_{ l}| = o_p(1), \end{eqnarray}$

$\begin{eqnarray} \sup\limits_{i\le n, j, \beta\in \tilde B_n(r)}|\mu_{ij}(\beta)-\mu_{ij}| = |\mu^{(1)}(X_{ij}^\tau\beta^*)|| X_{ij}^\tau(\beta-\beta_0)| = o_p(1), \end{eqnarray}$

$\begin{eqnarray} &&\sup\limits_{i\le n, \beta\in \tilde B_n(r)}\lambda_{\max}\left\{\mbox{diag}[ R_i^{-1}A_i^{-1/2}(\beta)(\mu_i-\mu_i(\beta))] \right\}^2\\ &\le &\sup\limits_{i\ge 1, \beta\in \tilde B_n(r)}\lambda_{\max}\left\{ R_i^{-1}A_i^{-1}(\beta) R_i^{-1}]\right\}\times\sup\limits_{i\le n, \beta\in \tilde B_n(r)}\sum\limits_{j = 1}^{m_i}|\mu_{ij}-\mu_{ij}(\beta)|^2\\ & = & o_p(1), \end{eqnarray}$

$$$\sup\limits_{i\ge 1, \beta\in \tilde B_n(r)}\lambda_{\max}\left\{A_i^{-1/2}(\beta)\ddot\mu_i(\beta)\right\}^2<\infty\quad a.s..$$$

$\begin{eqnarray} \lambda_k^\tau B^{[1]}_n(\beta)\lambda_l = o_p(1). \end{eqnarray}$

$$$\sup\limits_{i\le n, \beta\in \tilde B_n(r)}\lambda_{\max}\left\{A_i^{1/2}(\beta) R_i^{-1}[\mbox{diag}(\mu_i-\mu_i(\beta))]^2R_i^{-1}A_i^{1/2}(\beta) \right\} = o_p(1),$$$

$$$\sup\limits_{i\ge 1\beta\in \tilde B_n(r)}\lambda_{\max}\left\{A_i^{-3/2}(\beta)\ddot\mu_i(\beta)\right\}^2<\infty \quad a.s.$$$

$\begin{eqnarray} \lambda_k^\tau B^{[2]}_n(\beta)\lambda_l = o_p(1) . \end{eqnarray}$

$$$\bar{{\cal E}}^{[1]}_n(\beta) = \bar M_n^{-1}H_n^{1/2}{{\cal E}}^{[1]}_n(\beta)H_n^{1/2}(\bar M_n^{-1})^\tau,$$$

$$$\bar{{\cal E}}^{[2]}_n(\beta) = \bar M_n^{-1}H_n^{1/2}{{\cal E}}^{[2]}_n(\beta)H_n^{1/2}(\bar M_n^{-1})^\tau,$$$

$$$\lambda_k^\tau\bar{{\cal E}}^{[1]}_n(\beta)\lambda_l = \sum\limits_{i = 1}^nS_{ni}^{[1]}+\sum\limits_{i = 1}^nS_{ni}^{[3]}+\sum\limits_{i = 1}^nS_{ni}^{[5]},$$$

$$$\lambda_k^\tau\bar{{\cal E}}^{[2]}_n(\beta)\lambda_l = \sum\limits_{i = 1}^nS_{ni}^{[2]}+\sum\limits_{i = 1}^nS_{ni}^{[4]}+\sum\limits_{i = 1}^nS_{ni}^{[6]},$$$

$$$\max\limits_{i\le n}\|\lambda^\tau_l \bar M_n^{-1} X_i\|^2\le C \max\limits_{i\le n}\|F_n^{-1/2} X_i \|^2 = o_p(1),$$$

$$$\sum\limits_{i = 1}^n\|\lambda^\tau_k \bar M_n^{-1} X_i\|^2\le\sum\limits_{i = 1}^n\| \bar M_n^{-1} X_i\|^2 = O_p(1).$$$

$\begin{eqnarray} \sup\limits_{i\ge 1} \lambda_{\max}(R_i^{-1}\bar R_i R_i^{-1})<\infty, a.s., \quad\sup\limits_{i\ge 1}\lambda_{\max}(A_i^{-1}\ddot\mu^2_i))<\infty, a.s. \end{eqnarray}$

$\begin{eqnarray} \sum\limits_{i = 1}^nE[(S_{ni}^{[1]})^2|{\cal F}_{i-1}] & = &\frac{1}{4} \sum\limits_{i = 1}^n\lambda_k^\tau\bar M_n^{-1}X_iA_i^{-1/2} \ddot\mu_i\mbox{diag}(\lambda_l^\tau \bar M_n^{-1}X_i) R_i^{-1}\bar R_i R_i^{-1}{}\\ & &{\qquad}\ \times\mbox{diag}(\lambda_l^\tau\bar M_n^{-1}X_i)\ddot\mu_i A_i^{-1/2}X_i^\tau(\bar M_n^{-1})^\tau\lambda_k{}\\ &\le &\frac{1}{4}\max\limits_{1\le i\le n} \lambda_{\max}(R_i^{-1}\bar R_i R_i^{-1})\times o_p(1) \times\max\limits_{1\le i\le n}\lambda_{\max}(A_i^{-1}\ddot\mu^2_i) \times O_p(1){}\\ & = &o_p(1). \end{eqnarray}$

$\begin{eqnarray} \sum\limits_{i = 1}^nE[(S_{ni}^{[1]})^2(|S_{ni}^{[1]}|>\epsilon)|{\cal F}_{i-1}] = o_p(1). \end{eqnarray}$

$$$\sum\limits_{i = 1}^nS_{ni}^{[1]} = o_p(1).$$$

$$$\sum\limits_{i = 1}^nS_{ni}^{[2]} = o_p(1).$$$

$\begin{eqnarray} \sup\limits_{\beta\in \tilde B_n(r)}|\sum\limits_{i = 1}^nS_{ni}^{[3]} | &\le& \sum\limits_{i = 1}^n\sup\limits_{\beta\in \tilde B_n(r)}\|\frac{-1}{2}\lambda_k^\tau\bar M_n^{-1}X_i[A_i^{-1/2}(\beta) \ddot\mu_i(\beta) -A_i^{-1/2} \ddot\mu_i ]\\ &&{\qquad}{\qquad}{\quad}\times\mbox{diag}(X_i^\tau (\bar M_n^{-1})^\tau\lambda_l) R_i^{-1}A_i^{-1/2}(\beta)\|\| Y_i-\mu_i \|\\ &\le&\delta_n\times \sum\limits_{i = 1}^n\|\frac{-1}{2}\lambda_k^\tau\bar M_n^{-1}X_i\|\times\|X_i^\tau (\bar M_n^{-1})^\tau\lambda_l\| \times\| Y_i-\mu_i \|\\ &\le&\delta_n\times\sum\limits_{i = 1}^n\|\bar M_n^{-1}X_i\|^2 \times\| Y_i-\mu_i \|, \end{eqnarray}$

$\begin{eqnarray} \delta_n = o_p(1). \end{eqnarray}$

$$$\sum\limits_{i = 1}^n\|\bar M_n^{-1}X_i\|^2E[\|Y_i-\mu_i\||{\cal F}_{i-1}]\le \sup\limits_{i\ge 1}E[\|Y_i-\mu_i\||{\cal F}_{i-1}]\times\sum\limits_{i = 1}^n\|\bar M_n^{-1}X_i\|^2 = O_p(1).$$$

$e_i = \|Y_i-\mu_i\|-E[\|Y_i-\mu_i\||{\cal F}_{i-1}]$. 由假设(A3), (A6)(ⅰ), Jesen不等式, 式(5.19), (5.20)可得

$$$\sum\limits_{i = 1}^nE[\|\bar M_n^{-1}X_i\|^4e_i^2|{\cal F}_{i-1}] \le E[e_i^2|{\cal F}_{i-1}]\times\max\limits_{i\le n} \|\bar M_n^{-1}X_i\|^{2}\times\sum\limits_{i = 1}^n\|\bar M_n^{-1}X_i\|^2 = o_p(1),$$$

$\begin{eqnarray} && \sum\limits_{i = 1}^nE[\|\bar M_n^{-1}X_i\|^4e_i^2I(\|\bar M_n^{-1}X_i\|^2|e_i|>\epsilon)|{\cal F}_{i-1}] = o_p(1). \end{eqnarray}$

$\begin{eqnarray} \sum\limits_{i = 1}^n\|\bar M_n^{-1}X_i\|^2e_i = o_p(1). \end{eqnarray}$

$\begin{eqnarray} \sup\limits_{\beta\in \tilde B_n(r)}|\sum\limits_{i = 1}^nS_{ni}^{[3]} | = o_p(1). \end{eqnarray}$

$\begin{eqnarray} \sup\limits_{\beta\in \tilde B_n(r)}|\sum\limits_{i = 1}^nS_{ni}^{[4]} | = o_p(1), \quad \sup\limits_{\beta\in \tilde B_n(r)}|\sum\limits_{i = 1}^nS_{ni}^{[5]} | = o_p(1) , \quad \sup\limits_{\beta\in \tilde B_n(r)}|\sum\limits_{i = 1}^nS_{ni}^{[6]} | = o_p(1). \end{eqnarray}$

$$$\max\limits_{i\le n}\|a_{ni}\|\le \|\lambda^\tau\bar M_n^{-1} F_n^{1/2}\| \|\max\limits_{i\le n}\|F_n^{-1/2} X_i \|\|A_i^{1/2}R_i^{-1}\| = o_p(1),$$$

$$$\sum\limits_{i = 1}^n\|a_{ni}\|^2\le [\inf\limits_i\lambda_{\min}(\bar R_i)]^{-1}\sum\limits_{i = 1}^n\lambda^\tau\bar M_n^{-1}M_n(\bar M_n^{-1})^\tau\lambda^\tau = O_p(1).$$$

$$$\sum\limits_{i = 1}^nE[Z_{ni}^2(I(|Z_{ni}|>\epsilon))|{{\cal F}}_{i-1}]\le\sup\limits_{i\ge 1} E(\|Y^*_i\|^{\alpha_0}|{{\cal F}}_{i-1}) \max\limits_{i\le n}\|a_{ni}\|^{\alpha_0-2}\sum\limits_{i = 1}^n\|a_{ni}\|^2\to_p0.$$$

$$$\sum\limits_{i = 1}^nE[Z_{ni}^2|{{\cal F}}_{i-1}] = \lambda^\tau\bar M_n^{-1}M_n(\bar M_n^{-1})^\tau\lambda\to_p 1.$$$

$\begin{eqnarray} M_n^{-1/2}g_n(\beta_0) = [\bar M^{-1}_n M_n^{1/2}]^{-1} \times\bar M_n^{-1}g_n(\beta_0) \to_d N(0, I), \end{eqnarray}$

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